Exercises: Chain rule

Given the function $f(x) = (3x^2 - 5)^4$, identify the "outer" function and the "inner" function. Once identified, apply the chain rule to find its derivative $f'(x)$.

Calculate the derivative of the following radical function:

$g(x) = \sqrt{5x + 2}$

Use the structure of the chain rule for powers, remembering that $\sqrt{u} = u^{1/2}$.

In the marketing department of a fashion startup, they have estimated that the number of registered users on their website follows the function:

$U(t) = e^{0,5t + 2}$

where $t$ is the time in months. Find the expression for the instantaneous rate of growth of the users (that is, the derivative $U'(t)$).

Find the derivative of $h(x) = \ln(x^2 + 3x)$. To do so, remember that if $y = \ln(u)$, then $y' = \frac{u'}{u}$.

The chain rule tells us that if we have a composition of functions $y = f(g(x))$, its derivative is $y' = f'(g(x)) \cdot g'(x)$.

Apply this concept to differentiate the trigonometric function:

$f(x) = \sin(4x - 1)$

An electronics manufacturer determines that the total cost (in euros) of producing $x$ units of a new sensor is given by:

$C(x) = (x + 10)^{3}$

Calculate the marginal cost when $x$ units are produced (that is, find $C'(x)$).

Observe the following diagram representing the composition of two functions $u(x)$ and $v(x)$. If we know that $f(x) = (v \circ u)(x)$, use the information in the diagram to calculate $f'(x)$.

$u(x) = x^3 + 1 \quad \longrightarrow \quad v(u) = \sqrt{u}$

Calculate the derivative of the following composite polynomial functions, simplifying the result as much as possible:

1. a) $f(x) = (1 - x)^5$
2. b) $g(x) = (2x + 3)^2$

Marcos is preparing for the university entrance exam and needs to differentiate the function $f(x) = \sqrt{5x^2 - 2x + 1}$. In his notebook, he writes the following solution:

He argues that, according to the basic table of derivatives, "the derivative of the square root is always one divided by twice the root itself".

1. a) Identify the outer function $v(u)$ and the inner function $u(x)$ that compose $f(x) = v(u(x))$.
2. b) Explain, with reasons, the conceptual error made by Marcos when applying the chain rule and which factor he forgot.
3. c) Correct the calculation by writing the simplified expression of the derivative function $f'(x)$ and calculate the value of the slope of the tangent line to the graph of $f$ at the point with abscissa $x = 1$.

We want to calculate the derivative function of $f(x) = \ln\left(\frac{2x + 1}{x + 3}\right)$ for any value in the domain where $x > -0,5$. To do so, two different approaches are proposed:

• Method A (Direct chain rule): Consider $u(x) = \frac{2x + 1}{x + 3}$, compute the derivative of the quotient $u'(x)$, and then apply $(\ln u)' = \frac{u'}{u}$.

• Method B (Logarithm properties): First apply the property $\ln\left(\frac{A}{B}\right) = \ln A - \ln B$ to transform the function into a difference of two simple logarithms, and then differentiate term by term.

1. a) Work out the derivative using both methods and verify that the same simplified expression is obtained in both cases.
2. b) Compare the algebraic workload of each procedure and justify which of the two methods you would recommend to a classmate to minimize the risk of making calculation errors.

Calculate the derivative of the following composite polynomial function and find the gradient of the tangent line to its graph at the point with abscissa $x = 1$:

$f(x) = (x^3 - 2x^2 + 3)^5$

An electronic component manufacturing company has estimated that the total cost (in thousands of euros) of producing $x$ units of a new processor model is given by the function:

$C(x) = \sqrt{2x^2 + 5x + 100}$

Obtain the expression for the marginal cost $C'(x)$ using the chain rule and calculate this cost when exactly $10$ units are produced.

The number of followers of a well-known influencer on a social network, measured in thousands of people, follows a growth model given by the function:

$S(t) = 50 \cdot \ln(t^2 + 4t + 5)$

where $t$ is the time in months since the launch of her advertising campaign. The graph shows the function and the point corresponding to $t = 3$.

Calculate the growth rate of followers (the derivative) in the third month ($t = 3$).

Given the composite exponential function $g(x) = e^{x^2 - 4x + 1}$, determine the points on the curve where the tangent line is horizontal.

Find the derivative of the following function, which combines the quotient rule with the chain rule:

$h(x) = \frac{(3x - 1)^2}{x^2 + 1}$

Simplify the final result as much as possible.

The market value of a technology share (in euros) has fluctuated during an 8-hour trading session according to the function:

$V(t) = 10 + 2 \cdot \sqrt{(t - 4)^2 + 9}$

where $t$ is the time in hours ($0 \le t \le 8$). Determine the instantaneous rate of change of the share's value at exactly the sixth hour of the trading day.

Determine the value of the constant parameter $k$ in the function $f(x) = \ln(kx^2 + 1)$ knowing that the slope of the tangent line to the graph of the function at the point with abscissa $x = 1$ is equal to $1$.

Let $f$ and $g$ be two differentiable functions. The following values are known:

• $g(2) = 5$

• $g'(2) = 3$

• $f'(5) = -4$

Use the chain rule to calculate the value of the derivative of the composite function $H(x) = (f \circ g)(x)$ at the point $x = 2$, that is, find $H'(2)$.

An olive oil cooperative in Jaén models its total daily packaging cost $C$ (in hundreds of euros) as a function of the volume $q$ (in thousands of liters packaged) using the function:

During an 8-hour continuous workday ($0 \le t \le 8$, where $t$ is measured in hours), the cumulative volume follows the function:

1. a) Justify, using the chain rule, the economic meaning of the composite derivative $\frac{dC}{dt}$ and express it theoretically in terms of $C'(q)$ and $q'(t)$.
2. b) Obtain the analytical expression for the instantaneous rate of change of cost with respect to time, $\frac{dC}{dt}$.
3. c) Calculate the exact value of $\frac{dC}{dt}$ 2 hours after the start of the shift ($t = 2$) and interpret the result, stating its corresponding economic units.

An economic research service analyzes the evolution of a business confidence index $I(x)$, which depends on the year-on-year inflation rate $x$ (expressed as a percentage), modeled by the function:

Currently, the inflation rate in Spain stands at $6\,\%$ ($x = 6$).

1. a) Estimate: Without differentiating analytically, reason what the sign of $I'(6)$ must be taking into account the economic context (does business confidence increase or decrease in the face of an inflation uptick?). Based on the shape of the function, estimate with reasons whether the negative impact on confidence will be milder or sharper when inflation goes from $6\,\%$ to $7\,\%$ than when it goes from $1\,\%$ to $2\,\%$.
2. b) Calculate: Obtain the derivative function $I'(x)$ using the chain rule and find its exact value at $x = 6$.
3. c) Check: Find the tangent line to the curve at $x = 6$, use it to estimate the value of the index if inflation were to rise slightly to $6,2\,\%$, and evaluate whether the obtained result is consistent with your initial estimate.

Calculate the derivative of the following expression, simplifying the result as much as possible using trigonometric identities or properties of logarithms:

Determine the values of the parameters $a, b \in \mathbb{R}$ so that the following function is continuous and differentiable across its entire domain, paying special attention to the composition in the second branch:

A digital marketing agency has determined that the number of active users on a social platform, $U(t)$, measured in thousands, follows a function depending on advertising spend $x$ (in thousands of euros), which in turn varies over time $t$ (in months). The functions are:

Calculate the instantaneous rate of change of the number of users with respect to time when exactly $t = 2$ months have elapsed. Interpret whether user growth is accelerating or slowing down at that moment.

Find the equation of the tangent line to the graph of the function $g(x)$ at the point with abscissa $x = 1$. The function is defined by the following composite expression:

In a study of a monopolistic market, it has been found that the price of a product $p$ (in euros) and the quantity demanded $q$ (in units) are implicitly related by the following equation:

Knowing that total revenue is given by the function $I = p \cdot q$, use the chain rule and implicit differentiation to find the derivative of revenue with respect to price, $\frac{dI}{dp}$, when the price is $p = 4$ euros (consider only the positive value of $q$).

Study the existence of points with a horizontal tangent in the function $f(x) = (x^2 - 4)^2 \cdot e^{-x^2}$. To do so, obtain the first derivative by applying the chain rule and the product rule, and set it equal to zero to find all possible solutions in the real domain.

Given the function $h(x) = \log_2 \left( \sqrt[3]{x^3 + 3x^2} \right)$, calculate the value of its derivative at the point with abscissa $x = 1$. Note: Remember to apply the chain rule for logarithms in bases other than $e$.

Let $f$ and $g$ be two functions for which the following data are known:

• $g(1) = 2$ and $g'(1) = 3$

• $f(2) = -1$ and $f'(2) = 5$

If we define a new function $H(x) = \sqrt{f(g(x)) + 5}$, calculate the value of $H'(1)$ by rigorously applying the chain rule. Comment on what would happen to the derivative if the value inside the square root were negative.

Consider the composite function $H(x) = \ln(x^2 + bx + c)$, where $b$ and $c$ are real parameters. The following information about its behavior is available:

• The function $H(x)$ has a relative extremum at the point with abscissa $x = 2$.

• The tangent line to the graph of $H(x)$ at the point with abscissa $x = 1$ is parallel to the line with equation $y = -x + 4$.

1. a) Apply the chain rule to obtain the expression for $H'(x)$ in terms of the parameters $b$ and $c$.
2. b) Set up a system of equations from the given data and determine the values of $b$ and $c$ with reasons. Furthermore, prove that, with these coefficients, the argument of the logarithm is strictly positive for any real number, so that the domain of $H(x)$ is all of $\mathbb{R}$.
3. c) Classify the relative extremum located at $x = 2$ (indicating whether it is a relative maximum or minimum) by rigorously studying the sign of $H'(x)$ to the left and to the right of that point.

A financial audit models the net profitability of two business branches using two unknown differentiable functions, $f$ and $g$. The explicit algebraic formulas are not available for either of them, but the following data are precisely known:

• The tangent line to the graph of $g(x)$ at the point with abscissa $x = 3$ has the equation $y = -2x + 7$.

• For the function $f$, the following values have been recorded: $f(1) = 4$, $f'(1) = -3$, $f(3) = 2$, and $f'(3) = 5$.

From these data, two new composite financial indicators are defined:

1. a) Deduce with reasons, from the given tangent line, the numerical values of $g(3)$ and of the derivative $g'(3)$.
2. b) Apply the chain rule to calculate the equation of the tangent line to the graph of the function $h(x)$ at the point with abscissa $x = 3$. Explicitly justify which values from the provided information about $f$ you use and which you must discard, warning about the common mistake of confusing $f'(g(3))$ with $f'(3)$.
3. c) Calculate the instantaneous rate of change $k'(3)$ and interpret whether the indicator $k(x)$ is in an increasing or decreasing phase at that point.

Topic challenge. At what rate does our electricity consumption change?

1. Data collection: Visit the Red Eléctrica de España (REE) website and record the peninsular demand (in MW) for six different hours of yesterday. Enter the hour-demand pairs into GeoGebra or a spreadsheet and fit a polynomial function $D(t)$.

2. Differentiation and chain rule: Suppose that the associated wholesale cost follows the model $C(D) = \sqrt{2 + 0,15 \cdot D^2}$ (in thousands of euros). Apply differentiation rules and the chain rule to analytically find the function for the rate of change of cost with respect to time: $C'(t) = \frac{dC}{dD} \cdot D'(t)$.

3. Derivative at a point and interpretation: Calculate the derivative at two key moments: off-peak hours ($t = 4\text{ h}$) and peak hours ($t = 21\text{ h}$). Interpret the value and sign of $D'(t)$ and $C'(t)$ as instantaneous rates of change.

4. Final product: Write a brief report (one page) with the graph, the mathematical developments, and a reasoned proposal on how to adapt consumption habits according to the speed at which energy expenditure changes.

Assessment criteria:

• Proper collection and modelling of real data in GeoGebra or a spreadsheet.

• Correctness in calculating derivatives by appropriately applying the chain rule.

• Rigorous mathematical and socioeconomic interpretation of the derivative at a point.

• Clarity of conclusions and quality in the presentation of the report.